is positive the solution is no more unique. For Problem (REF )
the principal existence results are due to Escobar [1]}, [2]}, [3]},
Marques [4]}, [5]} and Almaraz [6]}. Recenlty Brendle and
Chen [7]} and Mayer and Ndiaye [8]} covered all the remaining
cases. Concerning compactness of solutions in manifold of positive
type the first result for (REF ) is given by Felli and
Ould Ahmedou which in [9]} proved compactness for scalar flat
manifolds not conformally equivalent to the closed disk, using the
Positive Mass Theorem. Compactness has been proved also for manifolds
whose trace-free part of the second fundamental form is non-zero everywhere
on the boundary by Almaraz [10]} for dimensions \(n\ge 7\)  and
recently extended to dimension \(n=5,6\)  by Kim, Musso and Wey [11]},
and to dimensions \(n=3\)  [12]} and \(n=4\)  [11]} without
any further assumption on the second fundamental form. When the trace-free
part of the second fundamental form is non-zero everywhere on the
boundary the boundary of the manifold is said non umbilic and
Almaraz exploited this condition to bypass the Positive Mass Theorem.
This strategy has been recently adapted to manifold with umbilic
boundary, that is when the tensor of the second funtamental form vanishes
on the boundary, by the authors, provided that the Weyl tensor is
always different from zero on the boundary and \(n\ge 6\)  in [14]}, [15]}.
